Granular Instrumental Variables: Estimation and Inference

Granular Instrumental Variables: Estimation and Inference

Introduction

Understanding how aggregate outcomes respond to shocks is a central objective in economics and finance. In many applications, researchers observe a large cross section of entities that are simultaneously exposed to a small number of common shocks. Examples include firms responding to aggregate demand conditions, financial institutions adjusting portfolios in response to market forces, and countries reacting to global macroeconomic shocks. A common feature of these environments is that the variables of interest are jointly determined in equilibrium, creating endogeneity problems that complicate identification and estimation of structural parameters.

Recently, gabaix2024granular proposed a novel identification strategy based on granular instrumental variables (GIVs). The approach builds on the insight from the granularity literature that when a small number of firms, industries, countries, investors, or borrowers account for a non-negligible share of aggregate activity, idiosyncratic shocks to these units may survive aggregation and influence aggregate outcomes.\footnote{ See, among others, gabaix2011granular, acemoglu2012network,di2014firms, baqaee2019macroeconomic,gaubert2021granular .} Exploiting this feature, GIV extracts the idiosyncratic component of observables after controlling for common factors and aggregates these components using size weights to construct instruments for causal parameters such as elasticities and multipliers. Under suitable restrictions on the covariance structure of the idiosyncratic shocks, the resulting instruments are orthogonal to equilibrium disturbances and can therefore identify structural parameters. Unlike earlier approaches that use idiosyncratic shocks to variables excluded from the estimating equation,\footnote{ See, for instance, leary2014peer,amiti2018much,amiti2019international. } GIV constructs instruments from the idiosyncratic component of the variables entering the estimating equation itself. As a result, the methodology does not rely on traditional excluded instruments, which are often difficult to justify or unavailable in practice.

Several recent papers extend the baseline GIV framework. banafti2022inferential study inference in large-$n$, large-$T$ settings with unknown factors and loadings. baumeister2023uncovering develop a likelihood-based approach, while qian2023heterogeneity allows for heterogeneous spillovers. GIV has also become an important identification device in empirical macroeconomics and finance, including applications to stock-market demand, exchange rates, bank lending, and asset pricing.\footnote{ Recent applications include galaasen2020granular, camanho2022global,ma2022mutual,dong2025fast; many others are discussed in gabaix2024granular.}

Despite its growing importance, several econometric questions remain unresolved. The central challenge is that valid GIV construction requires knowledge of the factor-loading space associated with latent aggregate shocks, which is rarely observed in practice. Existing implementations therefore estimate latent factors and construct GIVs in a second step, a strategy that typically relies on a large cross-sectional dimension and strong normalization assumptions. Moreover, little is known about the consequences of estimating the factor-loading space for identification, estimation, and inference.

This paper develops an estimation and inference framework for structural models identified by GIVs when the factor-loading space is unknown. Rather than estimating latent factors and constructing instruments in a second step, we show that the relevant GIV space can be recovered directly from the covariance structure of the observables. Specifically, the admissible GIVs are generated by the orthogonal complement of the factor-loading space, which can be identified from the eigenspace associated with the smallest eigenvalues of the covariance matrix. This insight transforms the construction of GIVs into a covariance-based problem and yields a feasible procedure for constructing instruments directly from the data. The resulting estimator remains valid even when the number of entities is fixed and therefore does not require the cross-sectional dimension to diverge with the sample size.

The characterization also provides a transparent identification strategy. We show that all admissible GIVs are generated by the orthogonal complement of the column space spanned by the factor loadings and the vector of ones. When the factor-loading space is unknown, the relevant orthogonal complement can be consistently recovered from the eigenspace associated with the smallest eigenvalues of the covariance matrix of the observables. This result yields a feasible GIV estimator and forms the basis for inference with estimated GIVs.

Building on this characterization, we establish consistency and asymptotic normality of the feasible GIV estimator and develop practical procedures for inference and specification testing. In particular, we derive feasible standard errors, establish the asymptotic validity of an over-identification $ J$-test when the GIVs are estimated, and propose a BIC-type criterion for determining the dimension of the factor-loading space. Monte Carlo simulations show that the feasible estimator performs similarly to an oracle estimator that knows the true factor-loading space and that the proposed inference procedures perform well in finite samples.

Beyond estimation and inference, the paper clarifies several identification issues that arise when factor loadings are unknown. We show that certain restrictions commonly interpreted as normalizations on the factor loadings instead impose substantive restrictions on the latent factors. We also examine the identification strategy in gabaix2024granular and show that the moment conditions in their Proposition 7 may fail to identify the structural parameters when factor loadings are unknown. These findings highlight the challenges of identification in the presence of latent aggregate shocks and motivate the alternative characterization developed in this paper. The proposed framework avoids these restrictions and extends naturally to settings with additional exogenous regressors, unbalanced panels, and heterogeneous demand elasticities.

Finally, we return to the estimation of the aggregate equity market multiplier in demand-based asset pricing gabaix2021search. This application is particularly relevant because the factor-loading space is unknown and only twelve investor sectors are available in the data. As a result, the large-cross-section justification underlying existing GIV procedures is difficult to invoke directly, making the setting a natural environment in which to assess the practical importance of estimating GIVs when factor loadings are unobserved. Applying our framework, we obtain estimates and conduct inference for the aggregate multiplier together with a formal specification test of the underlying GIV moment conditions. The empirical results provide evidence consistent with highly inelastic aggregate equity demand.

The remainder of the paper is organized as follows. Section (ref) introduces the main ideas in a simplified framework. Section (ref) develops the general model, establishes identification, and presents estimation and inference procedures based on estimated GIVs. Section (ref) studies identification when factor loadings are unknown, demonstrates a failure of identification in the moment conditions proposed by gabaix2024granular, and develops extensions to models with exogenous regressors, unbalanced panels, and heterogeneous demand elasticities. Section (ref) reports Monte Carlo evidence, Section (ref) presents an empirical application to the aggregate equity market multiplier, and Section (ref) concludes. Proofs and additional technical and empirical results are collected in the Online Appendix.

Notation. We use $K$ to denote a generic strictly positive constant that may vary from place to place but does not depend on the sample size $T$. We write $a\equiv b$ to indicate that $a$ is defined as $b$. For any positive integer $k$, let $\mathbf{I}_k$, $\mathbf{1}_k$, and $\mathbf{0}_k$ denote the $k\times k$ identity matrix, the $k\times1$ vector of ones, and the $k\times1$ vector of zeros, respectively. For any vector $x_t\in\mathbb{R}^n$ (possibly indexed by $t$) and any weight vector $\Greekmath 0121 \in\mathbb{R}^n$ satisfying $\Greekmath 0121 ^\top\mathbf{1}_n=1$, define the weighted average $x_{\Greekmath 0121 ,t}\equiv\Greekmath 0121 ^\top x_t$. For any matrix $A$, let $\func{col}(A)$ and $\func{rank}(A)$ denote its column space and rank, respectively. We use $\Vert A\Vert$ and $\Vert A\Vert_{\mathrm{o}}$ to denote the Frobenius norm and operator norm of $A$, respectively, and $M_A$ to denote the orthogonal projection matrix onto the orthogonal complement of $\func{col}(A)$. For any square matrix $A$, let $\Greekmath 011A _{\min}(A)$ and $\Greekmath 011A _{\max}(A)$ denote its smallest and largest eigenvalues, respectively. For any two matrices $A$ and $B$, let $\mathrm{diag}(A,B)$ denote the block-diagonal matrix with $A$ and $B$ on its main diagonal, and let $A\otimes B$ denote their Kronecker product. Finally, for any square matrix $A$, let $\func{vech}(A)$ denote the half-vectorization of $A$, obtained by stacking the elements of its lower triangular part (including the diagonal) column by column.

A Simplified Framework

We first illustrate the intuition of GIV using the following simplified model:

where $y_{i,t}$ denotes the log demand of entity $i\in \{1,\ldots,n\}$ at time $t$, and $p_{t}$ is the log price common to all entities. The latent variables $\Greekmath 0111 _{t}$ and $u_{i,t}$ with $\mathbb{E}[u_{i,t}]=0$ represent aggregate and idiosyncratic demand shocks, respectively. The parameter $\Greekmath 011E $ measures the demand elasticity. Equation ((ref)) describes the supply side, where $y_{S,t}\equiv S^{\top}y_{t}$ denotes aggregate demand, $ S\equiv(s_{i})_{i\leq n}$, and $y_{t}\equiv(y_{i,t})_{i\leq n}$, with $s_{i}$ denoting the market share of entity $i$. The term $\Greekmath 0122 _{t}$ is the supply shock, and $\Greekmath 0120 $ denotes the supply elasticity. Although stylized, this model is widely used in the macroeconomics and finance literature gabaix2021search,camanho2022global.

Let $u_{t}\equiv(u_{i,t})_{i\leq n}$. The following assumptions are maintained throughout this section:

where $\Greekmath 011B _{\Greekmath 0122 u}$ and $\Greekmath 011B _{\Greekmath 0111 u}$ are constants that need not be zero. These conditions ensure the validity of the GIVs constructed in the literature and considered in this section.\footnote{gabaix2024granular impose the stronger restrictions $\Greekmath 011B _{\Greekmath 0122 u}=0$ and $\Greekmath 011B _{\Greekmath 0111 u}=0$ for the identification of $\Greekmath 011E $ and $\Greekmath 0120 $ (see the first sentence of the paragraph containing their display (3)). As we show below, these restrictions are not necessary. Moreover, as shown in the next section, identification and estimation of $\Greekmath 011E $ and $\Greekmath 0120 $ do not require the distributions of the aggregate and idiosyncratic shocks to be time-invariant. In particular, their variances and covariances, such as $ \mathrm{Var}(u_{i,t})$ and $\mathrm{Cov}(\Greekmath 0111 _{t},u_{t})$, are allowed to vary over time.}

Following gabaix2024granular, we use the equally weighted average $ y_{e,t}\equiv e^{\top}y_{t}$, where $e\equiv n^{-1}\mathbf{1}_{n}$, to construct a GIV defined as

From the demand equation ((ref)), this GIV satisfies $z_{t}(e)=u_{t}^{\top}(S-e)$. Together with the first condition in ((ref)), this implies

Similarly,

The GIV $z_{t}(e)$ thus provides moment conditions ((ref)) and ( (ref)) for identifying and estimating the elasticities $\Greekmath 011E $ and $\Greekmath 0120 $.

The above identification strategy can be generalized to construct generic GIVs

where $a\in \mathbb{R}^{n}$ satisfies

Under these conditions, moment restrictions analogous to ((ref)) and ((ref)) can be constructed:

The GIV in ((ref)) corresponds to the special case $a=e$.

When $n>2$, there exist multiple vectors $a$ satisfying ((ref)). Therefore multiple GIVs are available and $\Greekmath 0120 $ and $ \Greekmath 011E $ become over-identified. This provides a natural motivation for using multiple GIVs both to improve efficiency and to conduct specification tests of instrument validity. We next characterize the resulting set of moment conditions and clarify its connection to the GIV-based approach.

The restrictions in ((ref)) imply the following moment conditions:

which together provide $n(n+3)/2$ moment conditions.\ To separate the parameters of interest from the nuisance parameters, let $ Q\equiv(q_{1},\ldots,q_{n})$ be an $n\times n$ orthonormal matrix with

where $\ell_{j}$ denotes the $j$th canonical basis vector of $\mathbb{R}^{n}$ . The following lemma provides a non-redundant representation of these moment conditions.

The moment conditions in ((ref)) and ((ref)) are equivalent to those in ((ref)) and ((ref)), and can be directly used within a GMM framework for estimation and inference of the unknown elasticities $\Greekmath 0120 $ and $\Greekmath 011E $.\footnote{ To establish this equivalence, note first that ((ref)) and ( (ref)) are constructed using GIVs of the form $q_{j}^{\top }y_{t}$ for $j\geq2$. For any $a\in \mathbb{R}^{n}$ satisfying ((ref)), the corresponding GIV is $(S-a)^{\top}y_{t}$, where $ (S-a)^{\top}\mathbf{1}_{n}=0$. Since $Q_{-1}$ spans the subspace orthogonal to $\mathbf{1}_{n}$, it follows that $S-a$ can be written as a linear combination of the columns of $Q_{-1}$. Hence, the moment conditions in ((ref)) and ((ref)) are implied by those in ((ref)) and ((ref)). Conversely, for each $j\geq2$ , the vector $S-q_{j}$ satisfies ((ref)), implying that ( (ref)) and ((ref)) are implied by ((ref)) and ((ref)).} In contrast, the restrictions in ((ref))--((ref)) and ((ref)) involve only nuisance parameters, namely $\Greekmath 011B _{u}^{2}$, $\mathbb{E} [\Greekmath 0111 _{t}^{2}]+2\Greekmath 011B _{\Greekmath 0111 u}$, and $\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u}$.

Specifically, the moment conditions in ((ref)) identify $ \Greekmath 011B _{u}^{2}$ and also yield additional restrictions that do not depend on unknown parameters. Conditional on $\Greekmath 011E $, $\Greekmath 0120 $, and $\Greekmath 011B _{u}^{2}$, the quantities $\mathbb{E}[\Greekmath 0111 _{t}^{2}]+2\Greekmath 011B _{\Greekmath 0111 u}$ and $\mathbb{E} [\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u}$ are just-identified by ( (ref)) and ((ref)), respectively. Since $ \Greekmath 011B _{u}^{2}$ is over-identified by ((ref)), jointly estimating $\Greekmath 011E $, $\Greekmath 0120 $, and $\Greekmath 011B _{u}^{2}$ using ((ref) ), ((ref)), and ((ref)) may yield more efficient estimators of $\Greekmath 011E $ and $\Greekmath 0120 $ than those based only on ((ref)) and ((ref)); see, for example, ackerberg2014asymptotic.

The analysis in this section relies on a simplified demand specification in which the aggregate shock $\Greekmath 0111 _{t}$ enters with a known and homogeneous loading across entities. In many applications, however, aggregate shocks may have heterogeneous effects that are not directly observed, giving rise to a more general factor structure. In the next section, we extend the GIV framework to this setting, where the demand equation includes unobserved factors with unknown loadings. This introduces new identification and estimation challenges, as the moment conditions derived above are no longer directly applicable when the factor loadings are unknown.

Granular IVs in a General Model

In this section, we study estimation and inference using GIVs in a more general model in which the demand equation ((ref)) incorporates a set of unobserved factors with unknown factor loadings.\footnote{ The model ((ref))-((ref)) can be further extended to include exogenous regressors in both the demand and supply equations without affecting the nature of the estimation and inference procedures proposed in this section; see Subsection (ref) for details.}\ Specifically, we consider

Here $\Greekmath 0111 _{t}$ denotes an $r\times1$ vector of unobserved factors, and $ \Greekmath 0115 $ is an $n\times r$ matrix of factor loadings. The vector $\Greekmath 0111 _{t}$ may also include a constant term, in which case the corresponding loading captures unobserved entity fixed effects. While the supply equation appears identical to ((ref)),\ we now define\

where $S_{t}\equiv(s_{i,t})_{i\leq n}$, and $s_{i,t}$ is nonnegative and predetermined at time $t$.\footnote{ Throughout this section, we assume that the number of entities $n$ is fixed over $t$. The identification strategy, as well as the estimation and inference procedures proposed in this section, also apply to settings in which $n$ varies over time; see Subsection (ref) for details.}

In contrast to the simplified model studied in the previous section, we now allow the factor loadings $\Greekmath 0115 $ to be unknown, which renders the earlier identification and estimation results inapplicable and constitutes the main challenge addressed in this section. One approach to handling the unobserved factors $\Greekmath 0111 _{t}$ is to estimate them from $y_{t}$ (after partialling out $ p_{t}$ and entity fixed effects) using principal component analysis (see, e.g., gabaix2021search and banafti2022inferential). However, as noted in the literature (see, e.g., bai2003inferential), the consistency of the estimated factors typically requires the number of entities $n$ to diverge, which stands in sharp contrast to most applications of GIVs, where the number of entities is relatively small.

Another approach, proposed in gabaix2024granular, attempts to identify $\Greekmath 0115 $ jointly with the other unknown parameters in the model under normalization restrictions and conditions similar to (but stronger than) Assumption (ref) below. However, as we show in Subsection (ref), their identification strategy fails to identify the factor loadings and may lead to invalid inference.

The method proposed in this section is based on an identification result that applies for any $n$, whether finite or diverging. Although we follow gabaix2024granular and construct our inference procedures under an asymptotic framework with fixed $n$, as discussed in Subsection (ref), the method can be straightforwardly extended to settings in which $n$ is large or even exceeds $T$.

Identification

In this subsection, we first establish identification of the demand and supply elasticities $\Greekmath 011E $ and $\Greekmath 0120 $ given $\Greekmath 0115 $, thereby extending the results in Lemma (ref). We then provide a constructive identification result for the orthogonal complement of $\func{col}((\mathbf{1}_{n}, \Greekmath 0115 ))$, which forms the basis for the estimation and inference procedures developed in the next subsection. We begin by stating the conditions required for identification.

Assumption (ref)(i)--(iii) generalize the conditions in ((ref)) by allowing the joint distribution of the demand shocks $ u_{t}$, the supply shocks $\Greekmath 0122 _{t}$, and the factors $\Greekmath 0111 _{t}$ to vary over time. Under these conditions, we obtain

which provide a total of $n(n+3)/2$ moment conditions for the unknown parameters $\Greekmath 011E $, $\Greekmath 0120 $, $\Greekmath 011B _{u,t}^{2}$, $\mathbb{E}[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]$, $\Gamma_{\Greekmath 0111 u,t}$, $\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]$, and $\Greekmath 011B _{\Greekmath 0122 u,t}$. Assumption (ref)(iv) is primarily imposed to ensure identification of $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$, whose orthogonal complement will be used to construct the GIVs.

We now reorganize the moment conditions ((ref))-((ref) ) according to their roles in identifying the different parameters.

Lemma (ref) shows that the key moment conditions for identifying $\Greekmath 011E $ and $\Greekmath 0120 $ are given by ((ref)) and ((ref)), which are constructed from the generalized GIVs $\bar{ \Greekmath 0115 }_{\bot}^{\top}y_{t}$. The moment conditions associated with the diagonal elements of

provide identifying restrictions for $T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}$. In contrast, the moment conditions in ((ref)), as well as the off-diagonal elements of the matrix above, do not involve any unknown parameters and are therefore redundant from an identification standpoint. Nevertheless, they may be useful for improving the efficiency of the GMM estimator and for testing specification assumptions, such as Assumption (ref)(i). Finally, ((ref)) and ((ref)) impose restrictions on the nuisance parameters $T^{-1}\sum_{t\leq T}\mathbb{E} [\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]$, $T^{-1}\sum_{t\leq T}\Gamma_{\Greekmath 0111 u,t}$, $ T^{-1}\sum_{t\leq T}\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]$, and $ T^{-1}\sum_{t\leq T}\Greekmath 011B _{\Greekmath 0122 u,t}$, conditional on the identification of $\Greekmath 011E $, $\Greekmath 0120 $, and $T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}$.

The moment conditions in ((ref)) and ((ref)) for the identification of $\Greekmath 011E $ and $\Greekmath 0120 $ rely on the generalized GIVs $ \bar{\Greekmath 0115 }_{\bot}^{\top}y_{t}$. These instruments are, however, infeasible in practice when the factor loading matrix $\Greekmath 0115 $ is unknown. We therefore next show how to identify the column space of $\bar{\Greekmath 0115 }$, which in turn determines the space spanned by $\bar{\Greekmath 0115 }_{\bot}$.

To this end, let $M_{\mathbf{1}_{n}}\equiv \mathbf{I}_{n}-n^{-1}\mathbf{1} _{n}\mathbf{1}_{n}^{\top}$ and define the demeaned variables

Applying $M_{\mathbf{1}_{n}}$ to both sides of ((ref)) yields

Combining ((ref)) with Assumption (ref), we obtain

Averaging ((ref)) over $t$ then yields

where

Here the matrix $\bar{\Sigma}_{\tilde{y}}$ is identified and can be consistently estimated.\ The following lemma shows that given the identification of $\bar{\Sigma}_{\tilde{y}}$, ((ref)) is sufficient to identify both $\bar{\Greekmath 011B }_{u}^{2}$ and the subspace orthogonal to the column space of $\bar{\Greekmath 0115 }$.

Lemma (ref) provides a constructive characterization of $ \bar{\Greekmath 0115 }_{\bot}$, which is central to the construction of generalized GIVs. In particular, $\bar{\Greekmath 0115 }_{\bot}$ can be recovered as the eigenspace associated with the smallest eigenvalue of $\bar{\Sigma}_{\tilde{y }}$ restricted to $\mathcal{B}_{\mathbf{1}_{n}}$. Intuitively, this corresponds to extracting directions of cross-sectional variation in $y_{t}$ that are orthogonal to both the common factor structure and the aggregate component spanned by $\mathbf{1}_{n}$.

The minimization problem in ((ref)), however, is defined over the constrained set $\mathcal{B}_{\mathbf{1}_{n}}$, which is not directly convenient for implementation. To facilitate computation, we next provide an equivalent representation that transforms this constrained problem into an unconstrained eigenvalue problem in $\mathbb{R}^{n-1}$.

The solutions to ((ref)) are given by the normalized eigenvectors associated with the smallest eigenvalue of the symmetric matrix $Q_{-1}^{\top}\bar{\Sigma}_{y}Q_{-1}$. By Lemma (ref) and Lemma (ref), these eigenvectors, after left multiplication by $Q_{-1}$, span the same subspace as $\bar{\Greekmath 0115 }_{\bot}$, thereby providing a feasible representation of the generalized GIVs. In practice, $ Q_{-1}^{\top }\bar{\Sigma}_{y}Q_{-1}$ can be consistently estimated by its sample analogue, so the GIVs can be implemented via standard eigenvalue decomposition without requiring knowledge of the factor loadings $\Greekmath 0115 $.

Estimation and inference with GIVs

Building on Lemmas (ref)--(ref) in the previous subsection, the moment conditions ((ref)) and ((ref)) for the identification and estimation of $\Greekmath 0112 \equiv(\Greekmath 011E ,\Greekmath 0120 )^{\top}$\ can now be written as

where

Here $A\equiv Q_{-1}A_{0}$, where $Q_{-1}$ is an $n\times(n-1)$ matrix defined in Lemma (ref), and $A_{0}$ is an $(n-1)\times(n-\bar{r} )$ matrix collecting the eigenvectors corresponding to the smallest $n-\bar{r }$ eigenvalues of

Since\ $A_{0}$ depends on the unknown population covariance matrix, the moment function $\bar{g}_{T}(\Greekmath 0112 ;A)$ is not directly feasible in practice.

To construct feasible moment conditions, we replace $A$ in ((ref) ) with $\hat{A}\equiv Q_{-1}\hat{A}_{0}$, where $\hat{A}_{0}$ collects the eigenvectors corresponding to the smallest $n-\bar{r}$ eigenvalues of

The GIV estimator is then defined as

where

and $W_{0,T}$ is a user-specified symmetric positive definite $2n\times2n$ matrix.

Since $\bar{g}_{T}(\Greekmath 0112 ;\hat{A})$ is linear in $\Greekmath 0112 $, the GIV estimator admits the closed-form representation

where $D_{j,T}(\hat{A})\equiv(\mathbf{I}_{2}\otimes \hat{A}^{\top})D_{j,T}$ for $j=1,2$, and

We next present sufficient conditions for establishing the asymptotic properties of $\hat{\Greekmath 0112 }(\hat{A})$ within the same framework as gabaix2024granular, where the number of entities $n$ is fixed and the number of observations $T$ (indexed by $t$) tends to infinity.\footnote{ The asymptotic properties of $\hat{\Greekmath 0112 }(\hat{A})$, as well as inference for the unknown parameter $\Greekmath 0112 $, can be extended to the case where both $n $ and $T$ diverge by applying techniques from the many-moments literature; see, for example, han2006gmm and newey2009generalized.} Let $ \{ \Greekmath 0116 _{j}\}_{j\leq n-1}$ denote the eigenvalues of $S_{y}$ arranged in increasing order, and let $A_{0,\bot}$ denote the matrix collecting the eigenvectors associated with $\{ \Greekmath 0116 _{j}\}_{n-\bar{r}+1\leq j\leq n-1}$.

Assumption (ref)(i) ensures that the demand and supply system admits a well-defined reduced form and, consequently, a unique equilibrium. Assumption (ref)(ii) guarantees that the population second moments of $(u_{t},\Greekmath 0122 _{t},\Greekmath 0111 _{t})$ are approximated by their sample counterparts at the rate $T^{-1/2}$. From Lemma (ref) , we have $\Greekmath 0116 _{j}=T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}$ for all $j\leq n- \bar{r}$. Therefore, Assumption (ref)(iii) imposes an eigenvalue gap condition on $S_{y}$, which is essential for consistent estimation of the eigenspace $\bar{\Greekmath 0115 }_{\bot}$. This condition can be verified under a lower bound condition on $\Greekmath 011A _{\min}((\mathbf{1}_{n},\Greekmath 0115 )^{\top }( \mathbf{1}_{n},\Greekmath 0115 ))$, or on $\Greekmath 011A _{\min}(\Greekmath 0115 ^{\top}\Greekmath 0115 )$ when $ \mathbf{1}_{n}\in \func{col}(\Greekmath 0115 )$; see Lemma (ref) in Online Appendix (ref) for details. Assumption (ref)(iii) also requires that the variance of the idiosyncratic demand shock be bounded away from zero, which is important for maintaining sufficient identification strength of the GIVs. Finally, Assumption (ref)(iv), together with (ref)(i), ensures that the second moments of $y_{t}$ and $ p_{t}$ are well defined.

Assumption (ref)(i) imposes a set of high-level moment conditions ensuring that sample averages involving the weighted aggregates, such as $S_{t}^{\top }u_{t}$, satisfy a standard $T^{-1/2}$ law of large numbers. In particular, it requires that interactions between market-share weights and the structural shocks $u_{t}$, $\Greekmath 0111 _{t}$ and $\Greekmath 0122 _{t}$\ exhibit sufficiently weak temporal dependence and possess finite second moments. This condition is analogous to Assumption (ref)(ii), and is implied by it when $S_{t}$ is time-invariant. More generally, both conditions can be verified under standard mixing or martingale difference assumptions. Assumption (ref)(ii) is a normalization condition requiring that the elements of $S_{t}$ sum to one.

Let $v_{t}\equiv y_{e,t}-\Greekmath 011E p_{t}$. For $b\in \{v,\Greekmath 0122 \}$, define

where

The random vectors $\Greekmath 0118 _{b,t}$, for $b\in \{v,\Greekmath 0122 \}$, represent the estimation errors in the moment conditions used to estimate $\Greekmath 011E $ and $\Greekmath 0120 $ , respectively. The first component, $y_{t}b_{t}-\mathbb{E}[y_{t}b_{t}]$, captures the sampling variation that would arise even if the\ factor loading matrix $\Greekmath 0115 $ were known. The second component reflects the additional estimation error induced by replacing $\bar{\Greekmath 0115 }_{\bot}$ with its estimator, and hence accounts for the impact of estimating the loading matrix on the moment conditions.

Assumption (ref)(i) concerns the asymptotic distribution of $ V^{-1/2}T^{-1/2}\sum_{t\leq T}\Greekmath 0118 _{t}$, which can be established via a central limit theorem. Here $V$ denotes the variance matrix of $ T^{-1/2}\sum_{t\leq T}\Greekmath 0118 _{t}$. Assumption (ref)(ii) ensures consistent estimation of the weighting matrix, while Assumption (ref)(iii) guarantees that the GIVs provide sufficient identification strength for $\Greekmath 011E $ and $\Greekmath 0120 $ to be $T^{1/2}$-estimable.\ The latter condition essentially requires that the weighted mean of the entities' market shares,\ $S_{u}\equiv T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}\mathbb{E} [S_{t}]$ does not lie in $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$. It can be verified under suitable primitive conditions; see Lemma (ref) in Online Appendix (ref) for details. Finally, Assumption (ref)(iv) requires the existence of a consistent estimator of $V$. \footnote{ A consistent estimator of $V$ can be constructed using the estimated shocks $ \hat{v}_{t}$ and $\hat{\Greekmath 0122 }_{t}$ obtained from the GIV estimator $ \hat{\Greekmath 0112 }(\hat{A})$ with identity weighting matrix $W_{0,T}=\mathbf{I} _{2n}$; see, for example, ((ref)) in the implementation algorithm in Online Appendix (ref). The consistency of this variance estimator is established in Theorem (ref) in the Online Appendix.}

To simplify the notation for the asymptotic variance of the GIV estimator, define

where $D_{1}(A)\equiv(\mathbf{I}_{2}\otimes A^{\top})D_{1},$

The following theorem establishes the asymptotic distribution of the GIV estimator.

Theorem (ref) shows that the asymptotic variance of the GIV estimator is minimized when $W_{0}=V$. Accordingly, the optimal weighting matrix can be obtained by setting $W_{0,T}=\hat{V}$ in ((ref)), which yields

Let $\hat{\Greekmath 0112 }^{\ast}(\hat{A})$ denote the corresponding optimally weighted GIV estimator. It then follows that

Standard errors for $\hat{\Greekmath 0112 }^{\ast}(\hat{A})$ can be constructed from the square roots of the diagonal elements of

whose validity follows from ((ref)).

Since there are $2(n-\bar{r})$ moment conditions for the identification and estimation of $\Greekmath 011E $ and $\Greekmath 0120 $, these parameters are over-identified whenever $n>\bar{r}+1$. In this case, the validity of the GIVs can be assessed using an over-identification test.

Theorem (ref) establishes the asymptotic distribution of the J-test statistic under the null hypothesis that the moment conditions ((ref)) are valid. When the GIVs are invalid, the power of the J-test follows from standard GMM arguments and is therefore omitted for brevity.

Estimating the number of GIVs

Construction of the GIVs requires knowledge of the rank $\bar{r}$ of the matrix $(\mathbf{1}_{n},\Greekmath 0115 )$, which may not be feasible in practice. In this subsection, we propose a Bayesian information criterion (BIC) for consistent estimation of $\bar{r}$. The construction is motivated by Lemma (ref) and Lemma (ref).

Specifically, let $\{ \hat{\Greekmath 0116 }_{j}\}_{j\leq n-1}$ denote the eigenvalues of $\hat{S}_{y}$ arranged in increasing order. Define the information criterion

for $j\in \mathcal{J}$, where $\mathcal{J}\equiv \{1,\ldots,n-1\}$. The estimator $\hat{r}$ of $\bar{r}$ is then given by

The consistency of $\hat{r}$ is established in Theorem (ref).

We conclude this subsection by providing intuition for the construction of ( (ref)). The criterion $\mathrm{BIC}_{T}(j)$ consists of two components. The first term, $T(n-j)^{-1}\sum_{s=1}^{n-j}(\hat{\Greekmath 0116 }_{s}-\hat{\Greekmath 0116 } _{1})^{2}/(2\hat{\Greekmath 0116 }_{s}^{2})$, is decreasing in $j$ and captures an over-fitting effect analogous to that in classical regression settings, where $j$ reflects the dimension or complexity of the model. The second term, $j\log(T)$, is strictly increasing in $j$ and serves as a penalty on model complexity. The estimator $\hat{r}$ in ((ref)) therefore balances the trade-off between goodness-of-fit and model complexity.

The eigenvalues $\{ \hat{\Greekmath 0116 }_{j}\}_{j\leq n-1}$ are $T^{1/2}$-consistent estimators of $\{ \Greekmath 0116 _{j}\}_{j\leq n-1}$ under Assumptions (ref) (i, ii). Since $\Greekmath 0116 _{s}=\bar{\Greekmath 011B }_{u}^{2}$ for $s\in \{1,\ldots,n-\bar{r} \}$, it follows that for $j\geq \bar{r}$,

Consequently, the penalty term $j\log(T)$ dominates the first term in ((ref)). Since $j\log(T)$ is strictly increasing in $j$, $\mathrm{BIC}_{T}(j)$ is asymptotically minimized at $\bar{r}$ over $j\in \{ \bar{r},\ldots,n-1\}$ . On the other hand, for $j<\bar{r}$, we have $n-j\geq n-\bar {r}+1$, so Assumption (ref)(iii) implies that $\Greekmath 0116 _{n-j}-\Greekmath 0116 _{1}$ is bounded away from zero. Combined with the $T^{-1/2}$ consistency of $\{ \hat{\Greekmath 0116 } _{j}\}_{j\leq n-1}$, this yields

which diverges at rate $T$. This term therefore dominates the penalty term of order $\log(T)$, implying that $\mathrm{BIC}_{T}(j)>\mathrm{BIC}_{T}(\bar{ r})$ wpa1 for all $j<\bar{r}$. Combining the two cases, $\mathrm{BIC}_{T}(j)$ is asymptotically minimized at $j=\bar{r}$ over $j\in \mathcal{J}$, which ensures the consistency of $\hat{r}$.

Extensions and Discussion

This section provides further discussion and extensions of the main results established in the previous section. First, we show that the identification strategy in gabaix2024granular may fail when the factor loadings are unknown, potentially leading to inconsistent estimation and invalid inference under standard GMM procedures. Second, we demonstrate that our estimation and inference procedures can be straightforwardly extended to settings with exogenous regressors in the demand and supply equations and to data with unbalanced features. The latter extension shows that our methods remain applicable even when the number of entities $n$ is large and may exceed $T$.

Identification failure of factor loadings in gabaix2024granular\

In this subsection, we show that the identification strategy in gabaix2024granular, in particular their Proposition 7, may fail when $ \Greekmath 0115 $ is unknown. gabaix2024granular impose a normalization on the factor loadings $\Greekmath 0115 $ by setting the loadings of the first factor $ \Greekmath 0111 _{1,t}$ to be $\mathbf{1}_{n}$, and the loadings of the remaining factors $\Greekmath 0111 _{2,t}$, denoted by $\Greekmath 0115 _{-1}$, to satisfy\footnote{ See the second paragraph above Proposition 7 in gabaix2024granular. In the same paragraph, they also impose the restriction that $\mathrm{Var} (\Greekmath 0115 _{-1}^{\top}y_{t})$ is diagonal with distinct diagonal entries\ ($ \Greekmath 0115 _{-1}^{\top}y_{t}$ here is equal to $n\check{\Greekmath 0111 }_{t}$ in their notation). As we show below, this additional restriction should be interpreted as an assumption on the latent factors rather than a normalization on the factor loadings. Moreover, imposing this restriction does not resolve the identification issue in their approach.}

Moreover, they assume\footnote{ See the second paragraph on page 2279, display (3), and Assumption 3 in gabaix2024granular.}

For notational simplicity, we abstract from exogenous regressors in both the demand and supply equations. The demand equation ((ref)) can then be written as

while the supply equation remains the same as ((ref)).

gabaix2024granular propose using both $\Greekmath 0115 _{\bot}^{\top}y_{t}$ and $\Greekmath 0115 _{-1}^{\top}y_{t}$ as IVs to construct moment conditions for identifying and estimating $\Greekmath 011E $, $\Greekmath 0120 $, and $\Greekmath 0115 _{-1}$.\footnote{ They correspond to $z_t(m^y)$ and $\check{\Greekmath 0111 }_t(m^y)$ in Proposition 4 of gabaix2024granular. } Because $\mathbf{1} _{n}^{\top}\Greekmath 0115 _{-1}=\mathbf{0}_{r-1}^{\top}$, one can partial out $ \Greekmath 0111 _{2,t}$ by premultiplying ((ref)) by $e^{\top}$, yielding

Moreover, because $n^{-1}\Greekmath 0115 _{-1}^{\top}\Greekmath 0115 _{-1}=\mathbf{I}_{r-1}$, premultiplying ((ref)) by $\Greekmath 0115 _{-1}^{\top}$, we also obtain

From Assumption (ref)(i) and ((ref)), it follows that the product of ((ref)) and the second term on the right of ((ref)) is such that

However, the product $\mathbb{E}[\Greekmath 0111 _{1,t}\Greekmath 0115 _{-1}^{\top}y_{t}]$ of the first term on the right of ((ref)) and ((ref)) may be nonzero due to possible correlation between $\Greekmath 0111 _{1,t}$ and $\Greekmath 0111 _{2,t}$. Therefore, $\Greekmath 0115 _{-1}^{\top}y_{t}$ cannot be directly used together with $ y_{e,t}-\Greekmath 011E p_{t}$ to form valid moment conditions. To address this issue, gabaix2024granular introduce the regression coefficient $b_{y}$ of $ \Greekmath 0111 _{1,t}$ on $y_{t}^{\top}\Greekmath 0115 _{-1}$ such that

Combining this with ((ref)) and ((ref)) yields

Since $y_{e,t}-\Greekmath 011E p_{t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{y}=u_{e,t}-u_{t}^{\top }\Greekmath 0115 _{-1}b_{y}+\Greekmath 0111 _{1,t}-n\Greekmath 0111 _{2,t}^{\top}b_{y}$, it follows from Assumption (ref)(i) and ((ref)) that

Together with ((ref)), this yields the moment conditions from the demand equation:

Similarly, $\Greekmath 0115 _{-1}^{\top}y_{t}$ cannot be directly used together with $ p_{t}-\Greekmath 0120 y_{S,t}$ to identify $\Greekmath 0120 $, because $\Greekmath 0115 _{-1}^{\top}y_{t}$ contains $\Greekmath 0111 _{2,t}$, which may be correlated with $\Greekmath 0122 _{t}$. gabaix2024granular therefore propose

where $b_{p}$ is defined by

Moreover, by Assumption (ref)(i) and ((ref)),

which provides additional moment conditions. Therefore, the moment conditions from the supply equation are

The moment conditions in ((ref)) and ((ref)) coincide with those in Proposition 4 of gabaix2024granular when additional exogenous variables are excluded. gabaix2024granular argue in their Proposition 4 that these conditions identify $\Greekmath 011E $, $\Greekmath 0120 $, $b_{y}$, and $ b_{p}$ when $\Greekmath 0115 _{-1}$ is known. When $\Greekmath 0115 _{-1}$ is unknown, they propose the additional moment conditions

which correspond to equation (50) in Proposition 7 of gabaix2024granular. To verify ((ref)), note that $M_{\mathbf{1} _{n}}\Greekmath 0115 _{-1}=\Greekmath 0115 _{-1}$ by ((ref)). Using ((ref)),

Hence, ((ref)) follows from Assumption (ref)(i).

Since ((ref)) provides $n(r-1)$ moment conditions for $n(r-1)$ unknown entries in $\Greekmath 0115 _{-1}$, it may seem to deliver exact identification.\footnote{ Indeed, gabaix2024granular state at the top of page 2294 that \textquotedblleft The new moment (50) identifies $\check {\Greekmath 0115 }$." In our notation, their moment (50) corresponds to ((ref)), while their $ \check{\Greekmath 0115 }$ is denoted here by $\Greekmath 0115 _{-1}$.} However, as shown in the lemma below, this is not the case: the restrictions in ((ref)) fail to uniquely identify $\Greekmath 0115 _{-1}$, even up to rotation.

Lemma (ref) shows that the moment condition ((ref)), together with the normalization in ((ref)), fails to identify $\func{ col}(\Greekmath 0115 _{-1})$. Indeed, ((ref)) and ((ref)) admit $ \binom{n-1}{r-1}$ different choices of $D_{J}$, whose column spaces are generally distinct. Moreover,

Therefore, if $\func{col}(D_{J})\neq \func{col}(\Greekmath 0115 _{-1})$, then the orthogonal complement of $(\mathbf{1}_{n},D_{J})$, denoted by $D_{J,\bot}$, need not be orthogonal to the true factor space $\func{col}(\Greekmath 0115 _{-1})$. In particular,

may hold. Consequently, the candidate GIVs $D_{J,\bot}^{\top}y_{t}$ may still contain components of the latent factors $\Greekmath 0111 _{t}$, since

As a result, moment conditions constructed from $D_{J,\bot}^{\top}y_{t}$ may fail to eliminate the latent factor component and therefore need not provide valid identifying restrictions for the structural parameters.

To make the identification problem more explicit, consider the special case $ r=n-1$. In this case, $\Greekmath 0115 _{\bot}$ is one-dimensional, and the moment conditions in ((ref)) and ((ref)) provide only exact identification for the unknown parameters $\Greekmath 011E $, $\Greekmath 0120 $, $b_{y}$, and $b_{p} $, given the true factor loading matrix $\Greekmath 0115 _{-1}$ and its orthogonal complement $\Greekmath 0115 _{\bot}$. Therefore, identification of these parameters ultimately relies on identification of $\func{col}(\Greekmath 0115 _{-1})$.

However, as discussed above, ((ref)) and ((ref)) admit $n-1$ different choices of $D_{J}$, and hence $n-1$ corresponding choices of $ D_{J,\bot}$. Since $\Greekmath 0115 _{\bot}$ is one-dimensional, at least $n-2$ of these choices satisfy

For such choices, the corresponding candidate GIVs $D_{J,\bot}^{\top}y_{t}$ retain latent factor components and therefore generally fail to identify the true elasticities $\Greekmath 011E $ and $\Greekmath 0120 $. Consequently, GMM estimation based on these invalid instruments would generally converge to pseudo-true values rather than the true structural parameters. Moreover, since different choices of $D_{J,\bot}$ generally lead to different pseudo-true values, the resulting limits need not even be uniquely determined.

Lemma (ref) establishes the non-identification of the factor loadings under the normalization in ((ref)). In addition to ((ref)), gabaix2024granular also assume that $\mathrm{Var} (\Greekmath 0115 _{-1}^{\top}y_{t})$ is a diagonal matrix with distinct diagonal entries.\footnote{ See the second paragraph above Proposition 7 in gabaix2024granular.} Under ((ref)) and ((ref)), however,

Hence, the diagonal structure imposed on $\mathrm{Var}(\Greekmath 0115 _{-1}^{\top }y_{t})$ amounts to additional restrictions on the latent factors, requiring that the components of $\Greekmath 0111 _{2,t}$ are uncorrelated and have distinct variances. Since $\mathbb{E}[y_{t}y_{t}^{\top}]$ is unknown, this condition should be viewed as an assumption on the latent factors rather than a restriction on the factor loadings.\footnote{ If one instead imposes the corresponding restriction on the sample second moment $T^{-1}\sum_{t\leq T}y_{t}y_{t}^{\top}$, then for a broad class of data-generating processes, the orthonormalized eigenvectors associated with any collection of $r-1$ eigenvalues may satisfy the empirical counterpart of this restriction in finite samples due to estimation error in $ T^{-1}\sum_{t\leq T}y_{t}y_{t}^{\top}$.}

Even after imposing this restriction together with ((ref)), the moment condition ((ref)) still fails to identify $\func{col} (\Greekmath 0115 _{-1})$. Indeed, by ((ref)), ((ref)), and ((ref)),

Under the additional assumption that $\mathbb{E}[\Greekmath 0111 _{2,t}\Greekmath 0111 _{2,t}^{\top}] $ is diagonal with distinct diagonal entries, the matrix $\mathbb{E}[\tilde { y}_{t}\tilde{y}_{t}^{\top}]$ has $r-1$ distinct eigenvalues larger than $ \Greekmath 011B _{u}^{2}$, while $\Greekmath 011B _{u}^{2}$ itself is an eigenvalue with multiplicity $n-r$.\ The eigenspace associated with $\Greekmath 011B _{u}^{2}$ is spanned by $\Greekmath 0115 _{\bot}$. Consequently, there are at least $(r-1)(n-r)+1$ such choices of $D_{J}$ with distinct column spaces.\footnote{ Note that one choice is given by $D_{J}=n^{-1/2}\Greekmath 0115 _{-1}$, while the remaining $(r-1)(n-r)$ choices are obtained by selecting $r-2$ columns from $ \Greekmath 0115 _{-1}$ columns and one column from $\Greekmath 0115 _{\bot}$.} Since these columns are orthonormal eigenvectors of $\mathbb{E}[\tilde{y}_{t}\tilde{y} _{t}^{\top}]$, it follows that

is diagonal with distinct diagonal entries. Since there exist $(r-1)(n-r)+1$ \ such choices of $D_{J}$ with distinct column spaces, this again shows that $\func{col}(\Greekmath 0115 _{-1})$ is not identified.

We conclude this subsection with a lemma establishing the rotational non-uniqueness of the factor loadings identified from ((ref)), ((ref))--((ref)) in the general case.

Lemma (ref) shows that the factor loadings $\Greekmath 0115 _{-1}$ and the regression coefficients $b_{y}$ and $b_{p}$ are, at best, identified up to an orthonormal rotation. This raises concerns for estimation and inference based on these moment conditions, since standard GMM procedures require uniqueness of the identified parameters and are therefore not directly applicable in this setting.

The non-identification issue here is more challenging to address than in our approach, because $\{b_{y},b_{p},\Greekmath 0115 _{-1},\Greekmath 0115 _{\perp}\}$ are jointly identified together with the demand and supply elasticities. This joint determination complicates both the computation of the GIV estimator and the analysis of its statistical properties. In contrast, our approach separates the identification of the factor loadings from that of the elasticity parameters. We then exploit the invariance properties of the GIV estimator and the $J$-test statistic to address the fact that the factor loadings are only identified up to rotation, thereby allowing standard GMM estimation and inference to remain valid.

Model with exogenous regressors

This subsection extends the model studied in the previous section by allowing for additional exogenous regressors in both the demand and supply equations in ((ref))--((ref)). Specifically, we consider

where $x_{t}\equiv(x_{1,t},\ldots,x_{n,t})^{\top}$ with $x_{i,t}\in \mathbb{R }^{d_{x}}$, and $w_{t}\in \mathbb{R}^{d_{w}}$ denote observed exogenous variables that have direct effects on demand and supply, respectively. The variables $x_{t}$ include both sector fixed effects and unit-level demand shifters, while $w_{t}$ captures aggregate supply shifters.\footnote{ Since $\Greekmath 0115 \Greekmath 0111 _{t}$ can be decomposed as $\Greekmath 0115 (\Greekmath 0111 _{t}-\mathbb{E} [\Greekmath 0111 _{t}])+\Greekmath 0115 \mathbb{E}[\Greekmath 0111 _{t}]$, and $\Greekmath 0115 \mathbb{E}[\Greekmath 0111 _{t}]$ can be absorbed into the sector fixed effects, we assume without loss of generality that $\mathbb{E}[\Greekmath 0111 _{t}]=\mathbf{0}_{r}$ throughout this subsection.} Under suitable exogeneity conditions, the main identification and estimation arguments continue to apply after partialling out $x_{t}$ from the demand equation ((ref)).\footnote{ When $x_{t}$ includes variables excluded from the supply equation, these may serve as IVs for identifying $\Greekmath 0120 $ in ((ref)) if they are uncorrelated with $\Greekmath 0122 _{t}$. Similarly, variables in $w_{t}$ excluded from the demand equation may identify $\Greekmath 011E $ in ((ref)) if they are uncorrelated with $\Greekmath 0111 _{t}$ and $u_{t}$. Although standard in the classical simultaneous equations literature, this strategy is not widely used in empirical applications of GIV. We therefore do not assume the existence or exogeneity of such excluded variables.}

Specifically, multiplying $M_{\mathbf{1}_{n}}$ on both sides of ((ref)) yields

where $\tilde{x}_{t}\equiv M_{\mathbf{1}_{n}}x_{t}$, and $\tilde{y}_{t}$, $ \tilde{\Greekmath 0115 }$, and $\tilde{u}_{t}$ are defined analogously; see ((ref)). Let $\tilde{y}_{t}^{\ast}\equiv M_{\mathbf{1}_{n}}y_{t}^{\ast} $, where $y_{t}^{\ast}\equiv y_{t}-x_{t}\Greekmath 010C $. Then the above equation can be written as

which takes a form similar to ((ref)).\footnote{ Since $w_{t}$ is invariant across $i$, it is automatically partialled out in $\tilde{y}$. Therefore, $\tilde{y}_{t}^{\ast}$ effectively partials out the exogenous regressors in both the demand and supply equations.} Therefore, we can apply Lemmas (ref) and (ref) in Subsection (ref), with the second moment matrix $\bar{ \Sigma}_{\tilde{y}}$ replaced by

to identify the subspace $\func{col}(\bar{\Greekmath 0115 }_{\bot})$, which is orthogonal to $(\mathbf{1}_{n},\Greekmath 0115 )$.

Given $\Greekmath 010C $, the GIVs and moment conditions can be constructed in the same way as in the previous section, with $y_{t}$ replaced by $y_{t}^{\ast}$. To proceed, we need to estimate $\Greekmath 010C $, which is required to construct an estimator for $y_{t}^{\ast}$. The unknown parameter $\Greekmath 010C $ is estimated by

If $x_{t}$ is exogenous, in the sense that it is uncorrelated with both $ \Greekmath 0111 _{t}$ and $u_{t}$, then standard least squares theory implies that $\hat{ \Greekmath 010C }$ is a $T^{1/2}$-consistent estimator of $\Greekmath 010C $. Given $\hat{\Greekmath 010C }$ , define

We then obtain $\hat{A}\equiv Q_{-1}\hat{A}_{0}$, where $\hat{A}_{0}$ collects the eigenvectors corresponding to the smallest\ $n-\bar{r}$ eigenvalues of

The moment conditions used to estimate the unknown parameter $\Greekmath 0112 \equiv(\Greekmath 011E ,\Greekmath 0120 ,\Greekmath 010D ^{\top})^{\top}$ are

The GIV estimator is defined as

where

and $W_{0,T}$ is a user-specified symmetric positive definite $ (2n+d_{w})\times(2n+d_{w})$ matrix. Since $\bar{g}_{T}(\Greekmath 0112 ;\hat{A},\hat{ \Greekmath 010C })$ is linear in $\Greekmath 0112 $, the GIV estimator takes the same form as in ( (ref)), with $D_{j,T}(\hat{A})$ redefined as

where

To conduct inference on $\Greekmath 0112 $ and test the validity of the moment conditions in ((ref)), one must account for the estimation error in $\hat{\Greekmath 010C }$, since it enters $\bar{g}_{T}(\Greekmath 0112 ;\hat{A},\hat{\Greekmath 010C })$ through both $\hat{y}_{t}^{\ast}$ and $\hat{A}$. Lemma (ref) in Online Appendix (ref) shows that the randomness introduced by the estimation error of $\hat{\Greekmath 010C }$ is of higher order. Therefore, the estimation error of $\hat{\Greekmath 010C }$ is asymptotically negligible and can be ignored. Consequently, the standard errors of $\hat{\Greekmath 0112 }(\hat{A})$ and the specification tests can be constructed in the same way as in the previous section. See Algorithm 1 in Online Appendix (ref) for details.

Model with unbalanced data structure

The data used to estimate the demand and supply equations have, thus far, been assumed to follow a balanced structure.\ For instance, $y_{i,t}$ denotes the demand of entity $i$ in period $t$, where $i\in \{1,\ldots,n\}$ and $t\in \{1,\ldots,T\}$. The analysis in the previous section assumes a balanced data structure, so that each time period is associated with the same number of entities. In practice, however, entry and exit lead to an unbalanced data structure. As we show below, the identification and estimation approach extends naturally to this setting, provided that the entry and exit decisions of entities are independent of their demand. \

Specifically, the demand equation in ((ref)) is generalized as

where $y_{t}\equiv(y_{i,t})_{i\leq n_{t}}$ and $n_{t}$ denotes the number of entities present in the market at time $t$. The factor-loading matrix $ \Greekmath 0115 $ is of dimension $n_t\times r$, and the idiosyncratic shock vector $u_t$ is of dimension $n_t\times1$. The supply equation in ((ref)) remains unchanged, with $y_{S,t}\equiv S_{t}^{\top}y_{t}$, where $S_{t}$ is an $n_{t}\times1$ vector of market shares.

To construct the GIV, consider a subsample of $n_{0}$ entities that are observed in all periods. Let $y_{t}^{0}$ denote the corresponding subvector of $y_{t}$. Their demand equation is

where $\Greekmath 0115 _{0}$ and $u_{t}^{0}$ are the associated submatrices of $ \Greekmath 0115 $ and $u_{t}$. Let $Q_{0,-1}\equiv(q_{0,2},\ldots,q_{0,n_{0}})\in \mathbb{R}^{n_{0}\times(n_{0}-1)}$ be defined analogously to ((ref) ) with $n$ replaced by $n_{0}$, and $\Greekmath 0115 _{0,\bot}\in \mathbb{R} ^{n_{0}\times(n_{0}-\bar{r}_{0})}$ denotes the orthogonal complement of $( \mathbf{1}_{n_{0}},\Greekmath 0115 _{0})$ where $\bar{r}_{0}\equiv \mathrm{rank}(( \mathbf{1}_{n_{0}},\Greekmath 0115 _{0}))$. Since $\Greekmath 0115 _{0,\bot}^{\top}y_{t}^{0}=\Greekmath 0115 _{0,\bot}^{\top}u_{t}^{0}$, Assumption (ref) implies

These imply $\Greekmath 0115 _{0,\bot}^{\top}y_{t}^{0}$ provides valid moment conditions

which identify $\Greekmath 011E $ and $\Greekmath 0120 $.

Lemmas (ref) and (ref) apply to this subsample. In particular, $\Greekmath 0115 _{0,\bot}$ can be consistently estimated (up to an orthonormal rotation) by $\hat{A}\equiv Q_{0,-1}\hat{A}_{0}$, where $\hat{A}_{0}$ collects the eigenvectors corresponding to the smallest $ n_{0}-\bar{r}_{0}$ eigenvalues of

The GIV estimator is defined analogously to ((ref)), with moment function

and a user-specified symmetric positive definite $2n_{0}\times2n_{0}$ weight matrix, and it admits an explicit form

where $D_{j,T}^{0}(\hat{A})\equiv(\mathbf{I}_{2}\otimes \hat{A} ^{\top})D_{j,T}^{0}$ for $j=1,2$, and

The asymptotic normality of the GIV estimator and the asymptotic distribution of the $J$-test statistic follow from the same arguments as in the proofs of Theorems (ref) and (ref), with the appropriate modifications to the $D_{1}$ and $V$ matrices. To conserve space, the implementation details of the GIV estimation and inference procedure are provided in Algorithm 2 in Online Appendix (ref).

Simulation Studies

We examine the finite-sample performance of the proposed GIV estimation and inference procedures through Monte Carlo experiments. Subsection (ref) describes the simulation design, and Subsection (ref) reports the results.

Simulation Setting\

We consider two simulation designs, a baseline design and an extended design, to investigate the finite-sample performance of the proposed GIV estimator, the associated inference procedures, and the specification test. The baseline design follows the model in ((ref))--((ref)), while the extended design augments the demand equation with three exogenous regressors.

Solving the demand-supply system in ((ref))--((ref)), with exogenous regressors in the demand equation but no additional exogenous variables in the supply equation, yields the following reduced-form expressions under the extended design:

To generate the simulated data, we first draw the demand and supply shocks $ \Greekmath 0111 _{t}$, $u_{t}$, and $\Greekmath 0122 _{t}$, the exogenous regressors $x_{t}$, and the market share vector $S_{t}$ conditional on the parameter values of $ \Greekmath 011E $, $\Greekmath 0120 $, $\Greekmath 010C $, and $\Greekmath 0115 $. These simulated values are then substituted into ((ref))--((ref)) to obtain $(p_{t},y_{t})$. This procedure generates the simulated observations $ \{y_{t},p_{t},x_{t},S_{t}\}$ for each period $t$.

The demand and supply shocks are mutually independent and i.i.d.\ across $t$ , with

where $\Sigma_{\Greekmath 0111 }\equiv((0.1)^{|i-j|})_{i,j\leq r+1}$, and $\Greekmath 011B _{\Greekmath 0122 }^{2}$ is set to $0.5$. The demand and supply elasticities are set to $\Greekmath 011E =-0.5$ and $\Greekmath 0120 =1.5$, respectively. The exogenous regressors $ \{x_{i,t}\}$ are generated i.i.d.\ from $N(\mathbf{0}_{3},\mathbf{I}_{3})$ across $i$ and $t$, independently of $(\Greekmath 0111 _{t}^{\top},u_{t}^{\top },\Greekmath 0122 _{t})^{\top}$. The market share vector $S_{t}$ is fixed across $ t$ and follows a Pareto rank-size specification:\ \ $s_{i}\propto \left( i/n\right) ^{-1/\Greekmath 0116 _{S}}$, with tail index $\Greekmath 0116 _{S}=0.2$, where the shares are normalized to satisfy $\sum_{i=1}^{n}s_{i}=1$. Such a power-law profile is consistent with the size distributions documented for industries, firms, and financial intermediaries gabaix2011granular,gabaix2024granular, and generates the concentrated cross-sectional structure under which granular variation is informative.

The factor loading matrix is specified as $\Greekmath 0115 =(\mathbf{1} _{n},\Greekmath 0115 _{-1})$, where $\Greekmath 0115 _{-1}$ is the $n\times r$ matrix of loadings on the $r$ non-aggregate latent factors. To construct $\Greekmath 0115 _{-1}$ , we first draw $nr$ independent $N(0,1)$ random variables to form an $ n\times r$ matrix $\Greekmath 0115 _{0}$. We then project $\Greekmath 0115 _{0}$ onto the orthogonal complement of $(\mathbf{1}_{n},S)$ and obtain a preliminary loading matrix $\tilde{\Greekmath 0115 }_{0}$ through the QR decomposition $M_{( \mathbf{1}_{n},S)}\Greekmath 0115 _{0}=\tilde{\Greekmath 0115 }_{0}R_{\Greekmath 0115 _{0}}$,\ where $ R_{\Greekmath 0115 _{0}}$ is an $r\times r$ upper triangular matrix and the columns of $\tilde{\Greekmath 0115 }_{0}$ are orthonormal. We then set $\Greekmath 0115 _{-1}=n^{1/2} \tilde{\Greekmath 0115 }_{0}$. By construction, the resulting loading matrix $ \Greekmath 0115 =(\mathbf{1}_{n},\Greekmath 0115 _{-1})$ has rank $\bar{r}=r+1$.

We set $\Greekmath 010C =\mathbf{0}_{d_{x}}$ in ((ref))--((ref)) for both designs. In the baseline design, $\Greekmath 010C $ is treated as known and the regressors $x_{t}$ are omitted. In the extended design, however, $\Greekmath 010C $ is estimated using the procedure described in Subsection (ref). Because $\Greekmath 010C =\mathbf{0}_{d_{x}}$, the regressors $x_{t}$ play no role in the data-generating process. Thus, any difference between the two designs reflects the additional estimation error associated with estimating $\Greekmath 010C $ and partialling out $x_{t}$.

We consider six combinations of the number of entities and the number of non-aggregate latent factors:

together with three sample sizes, $T\in \{150,300,450\}$.\ To evaluate the finite-sample performance of the GIV estimator, as well as the size properties of the associated inference and specification tests, we conduct $ 10{,}000$ Monte Carlo replications for each $(n,r,T)$ cell under each simulation design.

To assess the effect of estimation error arising from the recovery of the subspace orthogonal to $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$ on the performance of the GIV estimator, we consider an oracle GIV estimator constructed under the assumption that $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$ is known. Consequently, the matrix $A$, whose columns form a basis for the orthogonal complement of $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$ and are used to construct the GIVs, is treated as known.\footnote{ In the simulation, $A$ is constructed from the left singular vectors associated with the zero singular values in the singular value decomposition of $(\mathbf{1}_{n},\Greekmath 0115 )$.}\ The oracle GIV estimator therefore bypasses both the BIC step for estimating the number of factors and the estimation of the orthogonal complement of $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$. In the extended design, the oracle estimator additionally treats $\Greekmath 010C $ as known and partials out $x_{t}\Greekmath 010C $ using the true parameter value. By contrast, the feasible GIV estimator selects $\bar {r}$ using the BIC criterion in ( (ref))--((ref)), and then constructs the GIVs and the corresponding GIV estimator according to Algorithm 1 of Online Appendix (ref).

Both the oracle and feasible GIV estimators are evaluated using their finite-sample root mean squared errors (RMSEs), with the results reported in Table (ref) of the next subsection. We also investigate the empirical rejection probabilities of the two-sided tests of $H_{0}:\Greekmath 011E =-0.5$ and $H_{0}:\Greekmath 0120 =1.5$ at the $5\%$ significance level. The results are reported in Table (ref) of the next subsection. Inference is conducted using $t$-tests based on the standard error estimators described in Algorithm 1 of Online Appendix (ref), together with the asymptotic normality of the GIV estimators.

To evaluate the power of the $J$-test, we consider a controlled violation of the covariance restrictions underlying GIV validity while keeping the factor-loading matrix $\Greekmath 0115 =(\mathbf{1}_{n},\Greekmath 0115 _{-1})$, the structural parameters, and the marginal distributions of $u_{t}$, $\Greekmath 0111 _{t}$ , and $\Greekmath 0122 _{t}$ unchanged. Let $b_{n}=Ad_{n}$, where $d_{n}\in \mathbb{R}^{n-r-1}$ is the unit vector obtained by applying the Gram--Schmidt procedure to the first standard basis vector against $A^{\top}S $. For $\Greekmath 011A \in \lbrack0,1)$, we generate $u_{t}$ as

where $u_{t}^{\ast}\sim N(\mathbf{0}_{n},\mathbf{I}_{n})$ is generated together with $\Greekmath 0122 _{t}$ and $\Greekmath 0111 _{t}$ in the same way as $u_{t}$ in ((ref)).

By construction, $b_{n}\in \func{col}(A)$ implies $b_{n}^{\top }\mathbf{1} _{n}=0\ $and$\ b_{n}^{\top}\Greekmath 0115 _{-1}=\mathbf{0}_{r}^{\top}$, so $b_{n}$ is orthogonal to $\func{col}(\Greekmath 0115 )$. It is straightforward to verify that for any $\Greekmath 011A \in \lbrack0,1)$, the joint distribution of $ (u_{t},\Greekmath 0122 _{t})$ remains Gaussian with

When $\Greekmath 011A =0$, the design reduces to the correctly specified benchmark design used above to study the finite-sample properties of the GIV estimators and inference procedures. When $\Greekmath 011A >0$, however, $u_{t}$ and $ \Greekmath 0122 _{t}$ become correlated, with heterogeneous correlation patterns determined by $b_{n}$. As a result, the orthogonality conditions underlying the GIVs are violated, rendering the GIVs invalid. Moreover, the degree of misspecification increases linearly with $\Greekmath 011A $.

We use the six $(n,r)$ combinations in ((ref)) together with the three sample sizes $T$ considered above, vary $\Greekmath 011A $ over the grid $\Greekmath 011A _{j}=0.02j\ $for $j=0,\ldots,20$, and conduct $10{,}000$ simulation replications for each $(n,r,T,\Greekmath 011A )$ cell under each design. The significance level of the $J$-test is set at $0.05$, and the resulting empirical rejection probabilities are reported in Figure (ref) of the next subsection.

Simulation Results

Table (ref) shows that the RMSEs of both the oracle and feasible estimators of $\Greekmath 011E $ and $\Greekmath 0120 $ decline substantially as the sample size $T$ increases. For example, in the baseline design with $(n,r)=(5,1)$, the RMSE of the oracle estimator of $\Greekmath 011E $ decreases from $0.129$ at $T=150$, to $ 0.088$ at $T=300$, and further to $0.072$ at $T=450$. Similar improvements are observed across all configurations and for both structural parameters.

The feasible GIV estimator closely tracks the oracle estimator throughout the simulation designs. In most cases, the difference in RMSE between the oracle and feasible estimators is negligible. For instance, in the baseline design with $(n,r)=(10,5)$, the RMSEs of the feasible estimator $\hat{\Greekmath 011E } ^{\ast }(\hat{A})$ are $0.115$, $0.080$, and $0.064$ at $T=150$, $300$, and $ 450$, respectively, compared with the corresponding oracle RMSEs of $0.113$, $0.079$, and $0.064$. Similar qualitative patterns are observed for the GIV estimator of $\Greekmath 0120 $, as well as for both estimators in the extended design with exogenous regressors.

We next examine the size properties of the $t$-tests for $H_{0}:\Greekmath 011E =-0.5$ and $H_{0}:\Greekmath 0120 =1.5$ at the $5\%$ significance level. The results, reported in Table (ref), show that the empirical rejection probabilities approach the nominal level as the sample size $T$ increases. When the sample size is relatively small, i.e., $T=150$, the tests exhibit modest over-rejection, with the distortion becoming more pronounced in configurations involving a larger number of GIVs. For example, in the baseline design with $(n,r)=(8,3)$, where there are four GIVs and eight moment conditions, the empirical rejection probabilities of the $t$-tests based on the feasible GIV estimators for $\Greekmath 011E $ and $\Greekmath 0120 $ are $0.104$ and $ 0.073$, respectively, at $T=150$. By contrast, when $(n,r)=(8,5)$, where only two GIVs are available, the corresponding rejection probabilities are $ 0.075$ and $0.061$, respectively. Similar patterns are observed in the extended design.

The over-rejection observed in these $t$-tests does not appear to be primarily driven by estimation error in the number of factors or in the null space of $(\mathbf{1}_{n},\Greekmath 0115 )$, since the tests based on the oracle GIV estimators, which do not require estimation of these nuisance parameters, display similar finite-sample behavior. Instead, the over-rejection is likely related to the well-known many-moment bias in two-step GMM estimation, of which the GIV estimator is a special case; see, for example, HansenHeatonYaron1996 and newey2009generalized. Several approaches may help mitigate the resulting size distortion in small samples. For example, instead of estimating $\Greekmath 011E $ and $\Greekmath 0120 $ jointly, one may estimate them using two separate GMM procedures. This reduces the number of moment conditions used in each estimation problem, although it may sacrifice some of the efficiency gains from joint GMM estimation. Another possibility is to employ the continuously updated GMM estimator rather than the two-step GMM estimator. While this approach may improve finite-sample inference, it also introduces additional computational burden, since the continuously updated GMM estimator does not admit a closed-form solution.

Finally, we examine the performance of the $J$-test for assessing the validity of the moment conditions constructed using the GIVs. Table (ref) reports the empirical rejection probabilities at the $5\%$ nominal significance level under correct specification. The size behavior is broadly similar to that of the $t$-tests reported in Table (ref): the empirical rejection probabilities approach the nominal level as the sample size $T$ increases, while modest over-rejection is observed in small samples, particularly when the number of moment conditions is relatively large. For example, in the baseline design with $(n,r)=(10,5)$, the empirical rejection probabilities of the oracle and feasible $J$-tests are $ 0.069$ and $0.080$, respectively, at $T=150$. In the corresponding extended design, the rejection probabilities are $0.074$ and $0.083$, respectively. Similar patterns are observed for $(n,r)=(8,3)$, where the feasible rejection probabilities are $0.082$ in both the baseline and extended designs at $T=150$. As the sample size increases, the empirical rejection probabilities move steadily toward the nominal $5\%$ level across all configurations.

As discussed earlier for the $t$-tests, the observed small-sample over-rejection does not appear to be primarily driven by estimation error in the number of factors or in the null space of $(\mathbf{1}_{n},\Greekmath 0115 )$, since the oracle and feasible procedures display very similar finite-sample behavior. Instead, the distortion is likely related to the many-moment nature of the GMM problem.

We next examine the power of the feasible $J$-test under misspecification. Figure (ref) plots the empirical rejection probabilities of the $J$-test as a function of $\Greekmath 011A $, which controls the severity of the violation of the GIV moment conditions in the baseline design.\footnote{ The corresponding results for the extended design are reported in Online Appendix (ref) and exhibit similar qualitative patterns.} By construction, each power curve begins near the nominal $5\%$ level when $ \Greekmath 011A =0$. The rejection probabilities then increase monotonically with $\Greekmath 011A $ , and the power curves become substantially steeper as the sample size $T$ increases. For example, in the baseline design with $(n,r)=(10,7)$, the test achieves approximately $80\%$ power at $\Greekmath 011A \approx0.29$ when $T=150$, $ \Greekmath 011A \approx0.22$ when $T=300$, and $\Greekmath 011A \approx0.16$ when $T=450$. Similar patterns are observed across all configurations. The feasible $J$ -test exhibits good power in detecting moderate violations of the moment restrictions at empirically relevant sample sizes.

Empirical Application: Aggregate Market Multiplier

A central question in asset pricing is how strongly the aggregate stock market responds to shifts in investor demand for equities. This response is summarized by the aggregate market multiplier, denoted by $ \Greekmath 0114 \equiv-\Greekmath 011E ^{-1}$, where $\Greekmath 011E $ is the aggregate demand elasticity. Economically, $\Greekmath 0114 $ measures the change in aggregate equity value induced by a one-dollar demand shock. The aggregate demand elasticity has become a central object of interest in asset pricing and macro-finance because investor demand, portfolio reallocation, and market segmentation can have important effects on equilibrium asset prices; see, among others, piazzesi2007asset, koijen2019demand, and gabaix2021search. In frictionless benchmark models, aggregate demand is highly elastic, implying a multiplier close to zero. By contrast, the inelastic-markets hypothesis predicts a substantially larger value of $\Greekmath 0114 $.

The size of the aggregate multiplier remains controversial. Standard asset-pricing models imply a macro elasticity of roughly 10 to 20, corresponding to a multiplier of only 0.05 to 0.1.\footnote{ See Appendices F and I of gabaix2021search for computations of the macro elasticity implied by the model of lucas1978asset, the rare-disaster models of barro2006rare and gabaix2012variable , and the long-run risks model of bansal2004risks.} Yet empirical estimates of stock-level, factor-level, and aggregate demand elasticities generally point to substantially less elastic demand. For example, lou2012flow estimates a stock-level multiplier of about 1.2, pavlova2023benchmarking report multipliers between 0.3 and 0.5, and gabaix2021search report substantially larger multipliers at the aggregate level. These findings are consistent with the economic intuition that aggregate equity demand should be less elastic than demand for individual stocks, since stocks are closer substitutes for one another than for alternative asset classes such as bonds. At the same time, the estimated multipliers are an order of magnitude larger than those implied by standard asset-pricing models, posing a challenge for conventional theories of asset demand. Resolving this discrepancy requires credible identification of the aggregate demand elasticity, a key parameter for quantifying the effects of capital flows, institutional demand shocks, and policy interventions on asset prices.

Obtaining such identification is challenging because prices and quantities are jointly determined in equilibrium. Demand shocks affect prices, while prices simultaneously enter investors' demand equations, rendering simple regressions of demand on prices generally inconsistent. Building on the demand-based asset-pricing framework of koijen2019demand, gabaix2021search address this endogeneity problem by exploiting the latent-factor structure in investor demand and estimate aggregate multipliers ranging from 4.73 to 5.85, with a median estimate close to five. Their approach, however, relies on consistent estimation of the latent demand factors and therefore on asymptotic arguments in which the cross-sectional dimension diverges; see, for example, bai2003inferential. In this section, we revisit the aggregate multiplier using the GIV framework developed in Section (ref), which permits valid estimation and inference without requiring the number of sectors to grow with the sample size.

Following gabaix2021search, we study the aggregate multiplier through a demand system for U.S. equity holdings. Investors are grouped into $n$ equity-holding sectors.\ Let $\Delta q_{i,t}$ denote the fractional quarterly change in investor $i$'s equity holdings at quarter $t$, with its empirical counterpart defined in\ (ref) below; see Online Appendix (ref) for construction details. The demand equation is given by

where $\Delta p_{t}$ denotes the quarterly equity market return, $ \Greekmath 0111 _{t}\in \mathbb{R}^{r}$ represents latent aggregate demand factors, and $ u_{i,t}$ is an idiosyncratic demand shock. The parameter $\Greekmath 011E $ captures the aggregate demand elasticity and is the primary object of interest.\footnote{ Unlike gabaix2021search, we do not explicitly include aggregate macroeconomic variables, such as GDP growth, in ((ref)). Aggregate variables that enter the demand equation with homogeneous loadings across sectors are eliminated by the orthogonality condition defining the GIVs, since the instruments are constructed to be orthogonal to $\mathbf{1}_n$. Aggregate variables with heterogeneous loadings are absorbed into the latent factor component $\Greekmath 0115 \Greekmath 0111 _t$. Under the maintained assumption that the idiosyncratic shocks $u_{i,t}$ are orthogonal to these aggregate components, their omission does not affect identification, estimation, or inference for the demand elasticity $\Greekmath 011E $ and the aggregate multiplier $\Greekmath 0114 $. }

Because the aggregate supply of equity is approximately fixed in the short run, market clearing implies that size-weighted net demand equals zero:

where $\Delta q_{t}=(\Delta q_{i,t})_{i\le n}$ and $S_{t}$ denotes the vector of predetermined market shares. Combining (ref) and (ref) yields the equilibrium price equation

where $u_{S,t}\equiv S_{t}^{\top}u_{t}$. Thus, the aggregate multiplier $ \Greekmath 0114 $ measures the equilibrium price response to aggregate demand shocks, with a less elastic demand (smaller $|\Greekmath 011E |$) corresponding to a larger multiplier.

Our data are drawn from the Financial Accounts of the United States, Table L.224, which reports the equity holdings of major investor sectors. \footnote{ We use the June 2026 vintage of the Financial Accounts, in which corporate equity holdings by sector are reported in Table L.224.} Following gabaix2021search, our benchmark analysis uses the sample period 1993Q1--2018Q4 and includes twelve sectors that hold U.S. equities continuously throughout this period. Table (ref) in Online Appendix (ref) lists these sectors together with their average market shares. In the data, the fractional change in sector $i$'s equity holdings is measured as

where $w_{i,t}$ denotes the value of sector $i$'s equity holdings, $R_{t} $ is the gross capital-appreciation return on the aggregate stock market, and $\Delta p_{t}$ is measured by the quarterly simple return on the CRSP value-weighted index excluding dividends.\footnote{ Online Appendix (ref) details the construction of $\Delta q_{i,t}$ and $\Delta p_{t}$, and Online Appendix (ref) describes the sector classification and market-share weights $S_{t}$. Following gabaix2021search, pooled sector-level demand growth is winsorized at the 5th and 95th percentiles.} Beyond the benchmark sample, we consider an extended sample spanning 1988Q4--2025Q4, which is the longest period over which all twelve sectors are continuously observed.

For comparison, we also report estimates from the factor-residual IV (FIV) estimator of gabaix2021search, together with the OLS estimator. The FIV estimator constructs instruments from estimated idiosyncratic demand shocks obtained after removing observed and latent demand factors.\footnote{Online Appendix (ref) provides implementation details for the FIV estimator.} The OLS estimator is obtained from a regression of equally weighted demand, $n^{-1}\sum_{i\leq n}\Delta q_{i,t}$, on the market return $\Delta p_t$. Table (ref) reports the resulting estimates and standard errors.

Across all specifications, the OLS estimate of the aggregate multiplier exceeds its FIV and GIV counterparts. This pattern is consistent with the endogeneity problem discussed earlier. Positive demand shocks increase both holdings and prices, causing an uninstrumented regression to attribute part of the demand shock to the price response. As a result, the demand elasticity is biased toward zero and the implied multiplier is biased upward. By exploiting granular demand variation that is orthogonal to common demand factors, the GIV estimator corrects this source of bias.

When all twelve sectors are used to construct the GIVs, the estimated aggregate multiplier is $5.05$ in the benchmark sample 1993Q1--2018Q4 and $4.46$ in the extended sample 1988Q4--2025Q4, close to the corresponding FIV estimates. However, the over-identification test strongly rejects the associated moment restrictions in both samples, with $p$-values effectively equal to zero. One traditional interpretation of this result is that the instruments are invalid. Alternatively, following a perspective common in the treatment-effects literature, rejection of the over-identification test may reflect heterogeneity in the underlying causal parameters. From this perspective, the evidence suggests that the homogeneous-elasticity specification imposed on all twelve sectors is too restrictive and that demand elasticities may differ substantially across investor sectors.

To investigate this possibility, we restrict attention to the six largest sectors, which together account for more than 97% of total equity holdings in the sample: households, mutual funds and ETFs, the foreign sector, private pension funds, state and local pension funds, and life insurance companies. Relative to the twelve-sector specification, both the OLS and GIV estimates increase substantially. For example, the GIV estimate rises from $ 5.05$ to $8.70$ in the benchmark sample and from $4.46$ to $9.42$ in the extended sample. More importantly, the over-identification test no longer rejects, yielding $p$-values of $0.817$ and $0.594$ in the two samples. These findings suggest that the six largest sectors exhibit more homogeneous demand behavior and therefore provide a more credible basis for estimating a common aggregate demand elasticity and the corresponding market multiplier.

The comparison between the FIV and GIV estimators highlights the importance of the fixed-$n$ approach. When all twelve sectors are included, the two estimators deliver qualitatively similar conclusions. The FIV estimates of the aggregate multiplier are $4.42$ in the benchmark sample and $4.39$ in the extended sample, close to the corresponding GIV estimates of $5.05$ and $ 4.46$, respectively, and broadly consistent with the estimates reported by gabaix2021search. Both instrumental-variable estimators also yield smaller multipliers than OLS, as expected from the endogeneity bias discussed above.

The contrast becomes much sharper when attention is restricted to the six-sector granular core. In this case, the FIV estimator produces negative multiplier estimates of $-7.13$ and $-2.15$, whereas the GIV estimator yields stable and economically meaningful estimates of $8.70$ and $9.42$. This divergence reflects the different identification strategies underlying the two procedures. The FIV estimator constructs instruments from estimated idiosyncratic demand shocks obtained after removing latent factors through principal-components analysis and therefore relies on consistent estimation of those factors. With only six sectors, the cross-sectional dimension is too small for this approach to be reliable. By contrast, the fixed-$n$ GIV estimator does not require consistent estimation of latent factors and remains valid when the number of sectors is small. The resulting GIV estimates are therefore more credible for the six-sector specification, which is favored by the over-identification test and appears consistent with the homogeneous-elasticity restriction underlying the aggregate multiplier.

Table (ref) sheds light on the source of the rejection of the twelve-sector specification. Using the GIVs constructed from the six-sector core, we estimate sector-specific demand elasticities, $\Greekmath 011E _{j}$, and the corresponding implied market multipliers, $\Greekmath 0114 _{j}\equiv-\Greekmath 011E _{j}^{-1}$, for the remaining sectors. Several sectors exhibit substantially smaller multipliers than the six-sector core. Property and casualty insurers and state and local governments have multipliers between $3.5$ and $4.0$, less than half the core estimate. Closed-end funds have intermediate multipliers of roughly $6.5$, while banks exhibit larger and less precisely estimated multipliers. For all of these sectors, the $J$-test provides little evidence against the validity of the moment conditions constructed from the six-sector core.

The two remaining sectors, federal government retirement funds and broker-dealers, exhibit markedly different behavior. Their estimated multipliers are negative in both samples, implying non-positive estimates of the corresponding demand elasticities. Moreover, these estimates are highly imprecise and economically difficult to interpret. More importantly, the $J$ -test rejects the validity of the moment conditions for federal government retirement funds in both samples. For broker-dealers, the $J$-test is close to rejection at the 5% level in the shorter sample ($p$-value $=0.064$) and is strongly rejected in the longer sample ($p$-value $=0.011$). Taken together, these findings suggest that the moment conditions constructed from the six-sector core do not provide valid identifying restrictions for these two sectors. Consequently, the corresponding multiplier estimates should not be given an economic interpretation.

Taken together, Tables (ref) and (ref) provide strong evidence that demand elasticities differ substantially across investor sectors. Among the ten sectors for which the $J$-test does not reject (treating broker-dealers as invalid given the decisive rejection in the longer sample and the borderline $p$-value of $0.064$ in the shorter sample), the estimated multipliers range from approximately $3.6$ to $10.7$, with the upper end reflecting the imprecisely estimated bank sector. This heterogeneity explains both the rejection of the twelve-sector specification and the substantially smaller multiplier obtained when all sectors are pooled together. By contrast, the six-sector core appears considerably more homogeneous and yields a stable aggregate multiplier of roughly nine across both sample periods. This estimate is an order of magnitude larger than the frictionless benchmark multiplier of $0.05$ to $0.1$, providing evidence consistent with the inelastic-markets hypothesis.

Conclusion

This paper develops an estimation and inference framework for structural models identified by granular instrumental variables (GIVs). The key insight is that, under suitable cross-sectional restrictions on the idiosyncratic shocks, valid GIVs are characterized by the orthogonal complement of the factor-loading space associated with latent aggregate shocks. This characterization provides a transparent foundation for GIV-based identification and allows demand and supply elasticities to be estimated without conventional excluded instruments or direct observation of the latent factors.

A central contribution of the paper is to show that the relevant orthogonal complement can be identified and consistently estimated directly from the covariance structure of the observables, without first estimating the latent factors themselves. As a result, the proposed framework remains applicable even when the number of entities is fixed and does not require the cross-sectional dimension to diverge with the sample size. Building on this result, we develop feasible procedures for estimation, inference, and specification testing based on estimated GIVs. Monte Carlo evidence shows that the feasible estimator performs similarly to an oracle procedure that knows the true factor-loading space, while the empirical application yields evidence consistent with highly inelastic aggregate equity demand.

The analysis also clarifies several identification issues that arise when factor loadings are unknown. In particular, we show that certain restrictions commonly imposed in existing implementations of the GIV methodology should be interpreted as substantive assumptions on the latent factors rather than innocuous normalizations. We further show that identification can fail for existing moment-condition approaches when the factor-loading space is unknown. The framework developed here avoids these difficulties by focusing directly on the geometry of the factor-loading space and exploiting the eigenspace structure of the covariance matrix of the observables.

More broadly, the results demonstrate that cross-sectional heterogeneity can be used not only to construct granular instruments but also to conduct valid estimation, inference, and specification testing in models with latent aggregate shocks. We hope that the framework developed in this paper will facilitate the use of GIV methods in empirical work and provide a foundation for future research in settings where latent aggregate forces and granular heterogeneity interact.

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